basisgrid

Basisgrid is a grid structure defined by a basis of a vector space. It represents space by a set of grid points obtained as integer linear combinations of the basis vectors. Given a basis B = {b1,...,bk} in R^n, the basisgrid consists of points x = sum_i c_i b_i where c_i are integers within a chosen range. The basis matrix B = [b1 ... bk] maps grid coordinates c in Z^k to Cartesian coordinates x = B c. If the basis vectors form a full rank matrix, the set of all integer combinations yields a lattice; restricting coefficients to a finite window yields a finite basisgrid used for discretization.

Basisgrid generalizes rectangular grids by allowing oblique or skewed grids. Spacing and cell shapes depend on B and the selection of coefficient ranges. The grid inherits linear structure: affine transformations of the basis correspond to simple transformations of coordinates. Inverse mappings exist if B is invertible (i.e., k = n and det(B) ≠ 0).

Applications appear in numerical methods, computer graphics, and geographic information systems. In finite element analysis, basisgrids enable non-orthogonal meshes aligned with problem features. In texture sampling and rendering, oblique grids can reduce distortion. In signal processing on non-Cartesian domains, basisgrid supports irregular sampling.

Examples: the standard square lattice is obtained from the standard basis of R^2. A hexagonal lattice arises from basis vectors (1,0) and (1/2, sqrt(3)/2).

See also: lattice, coordinate system, basis (linear algebra), mesh grid, finite element method.